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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an associative algebra ''A'' is an
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
with compatible operations of addition, multiplication (assumed to be
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
), and a
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
multiplication, or, equivalently, an associative algebra that is also a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''R'', instead of a field: An ''R''-algebra is an ''R''-module with an associative ''R''-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if ''S'' is any ring with center ''C'', then ''S'' is an associative ''C''-algebra.


Definition

Let ''R'' be a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
(so ''R'' could be a field). An associative ''R''-algebra (or more simply, an ''R''-algebra) is a ring that is also an ''R''-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
satisfies :r\cdot(xy) = (r\cdot x)y = x(r\cdot y) for all ''r'' in ''R'' and ''x'', ''y'' in the algebra. (This definition implies that the algebra is unital, since rings are supposed to have a
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.) Equivalently, an associative algebra ''A'' is a ring together with a ring homomorphism from ''R'' to the center of ''A''. If ''f'' is such a homomorphism, the scalar multiplication is (r,x)\mapsto f(r)x (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by r\mapsto r\cdot 1_A (See also below). Every ring is an associative \mathbb Z-algebra, where \mathbb Z denotes the ring of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. A is an associative algebra that is also a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
.


As a monoid object in the category of modules

The definition is equivalent to saying that a unital associative ''R''-algebra is a monoid object in ''R''-Mod (the
monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
of ''R''-modules). By definition, a ring is a monoid object in the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of ...
; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the
category of modules In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ri ...
. Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra ''A''. For example, the associativity can be expressed as follows. By the universal property of a
tensor product of modules In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produc ...
, the multiplication (the ''R''-bilinear map) corresponds to a unique ''R''-linear map :m: A \otimes_R A \to A. The associativity then refers to the identity: :m \circ ( \otimes m) = m \circ (m \otimes \operatorname).


From ring homomorphisms

An associative algebra amounts to a ring homomorphism whose image lies in the center. Indeed, starting with a ring ''A'' and a ring homomorphism \eta\colon R \to A whose image lies in the center of ''A'', we can make ''A'' an ''R''-algebra by defining :r\cdot x = \eta(r)x for all ''r'' ∈ ''R'' and ''x'' ∈ ''A''. If ''A'' is an ''R''-algebra, taking ''x'' = 1, the same formula in turn defines a ring homomorphism \eta\colon R \to A whose image lies in the center. If a ring is commutative then it equals its center, so that a commutative ''R''-algebra can be defined simply as a commutative ring ''A'' together with a commutative ring homomorphism \eta\colon R \to A. The ring homomorphism ''η'' appearing in the above is often called a
structure map In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A ...
. In the commutative case, one can consider the category whose objects are ring homomorphisms ''R'' → ''A''; i.e., commutative ''R''-algebras and whose morphisms are ring homomorphisms ''A'' → ''A'' that are under ''R''; i.e., ''R'' → ''A'' → ''A'' is ''R'' → ''A'' (i.e., the
coslice category In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...
of the category of commutative rings under ''R''.) The
prime spectrum In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
functor Spec then determines an anti-equivalence of this category to the category of
affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
s over Spec ''R''. How to weaken the commutativity assumption is a subject matter of
noncommutative algebraic geometry Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as ge ...
and, more recently, of
derived algebraic geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutativ ...
. See also:
generic matrix ring In algebra, a generic matrix ring is a sort of a universal matrix ring. Definition We denote by F_n a generic matrix ring of size ''n'' with variables X_1, \dots X_m. It is characterized by the universal property: given a commutative ring ''R'' a ...
.


Algebra homomorphisms

A
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
between two ''R''-algebras is an ''R''-linear ring homomorphism. Explicitly, \varphi : A_1 \to A_2 is an associative algebra homomorphism if :\begin \varphi(r \cdot x) &= r \cdot \varphi(x) \\ \varphi(x + y) &= \varphi(x) + \varphi(y) \\ \varphi(xy) &= \varphi(x)\varphi(y) \\ \varphi(1) &= 1 \end The class of all ''R''-algebras together with algebra homomorphisms between them form a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
, sometimes denoted ''R''-Alg. The
subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of commutative ''R''-algebras can be characterized as the
coslice category In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...
''R''/CRing where CRing is the category of commutative rings.


Examples

The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.


Algebra

*Any ring ''A'' can be considered as a Z-algebra. The unique ring homomorphism from Z to ''A'' is determined by the fact that it must send 1 to the identity in ''A''. Therefore, rings and Z-algebras are equivalent concepts, in the same way that
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s and Z-modules are equivalent. *Any ring of characteristic ''n'' is a (Z/''n''Z)-algebra in the same way. *Given an ''R''-module ''M'', the
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
of ''M'', denoted End''R''(''M'') is an ''R''-algebra by defining (''r''·φ)(''x'') = ''r''·φ(''x''). *Any ring of matrices with coefficients in a commutative ring ''R'' forms an ''R''-algebra under matrix addition and multiplication. This coincides with the previous example when ''M'' is a finitely-generated, free ''R''-module. **In particular, the square ''n''-by-''n'' matrices with entries from the field ''K'' form an associative algebra over ''K''. * The
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s form a 2-dimensional commutative algebra over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. * The
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions). * The
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s with real coefficients form a commutative algebra over the reals. * Every
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''R'' 'x''1, ..., ''xn''is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set . * The free ''R''-algebra on a set ''E'' is an algebra of "polynomials" with coefficients in ''R'' and noncommuting indeterminates taken from the set ''E''. * The tensor algebra of an ''R''-module is naturally an associative ''R''-algebra. The same is true for quotients such as the exterior and
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
s. Categorically speaking, the
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
that maps an ''R''-module to its tensor algebra is left adjoint to the functor that sends an ''R''-algebra to its underlying ''R''-module (forgetting the multiplicative structure). *The following ring is used in the theory of λ-rings. Given a commutative ring ''A'', let G(A) = 1 + tA ![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t! the set of formal power series with constant term 1. It is an abelian group with the group operation that is the multiplication of power series. It is then a ring with the multiplication, denoted by \circ, such that (1 + at) \circ (1 + bt) = 1 + abt, determined by this condition and the ring axioms. The additive identity is 1 and the multiplicative identity is 1 + t. Then A has a canonical structure of a G(A)-algebra given by the ring homomorphism \begin G(A) \to A \\ 1 + \sum_ a_i t^i \mapsto a_1 \end On the other hand, if ''A'' is a λ-ring, then there is a ring homomorphism \begin A \to G(A) \\ a \mapsto 1 + \sum_ \lambda^i(a)t^i \end giving G(A) a structure of an ''A''-algebra.


Representation theory

* The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra. * If ''G'' is a group and ''R'' is a commutative ring, the set of all functions from ''G'' to ''R'' with finite support form an ''R''-algebra with the convolution as multiplication. It is called the group algebra of ''G''. The construction is the starting point for the application to the study of (discrete) groups. * If ''G'' is an algebraic group (e.g., semisimple
complex Lie group In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
), then the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal ...
of ''G'' is the
Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Swe ...
''A'' corresponding to ''G''. Many structures of ''G'' translate to those of ''A''. * A
quiver algebra In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation  of a quiver assigns a vector space&nb ...
(or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.


Analysis

* Given any
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
''X'', the
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s ''A'' : ''X'' → ''X'' form an associative algebra (using composition of operators as multiplication); this is a Banach algebra. * Given any
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'', the continuous real- or complex-valued functions on ''X'' form a real or complex associative algebra; here the functions are added and multiplied pointwise. * The set of semimartingales defined on the
filtered probability space Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
(Ω, ''F'', (''Ft'')''t'' ≥ 0, P) forms a ring under
stochastic integration Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created a ...
. * The Weyl algebra * An Azumaya algebra


Geometry and combinatorics

* The
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
s, which are useful in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
. * Incidence algebras of locally finite
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s are associative algebras considered in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
. * The partition algebra and its subalgebras, including the Brauer algebra and the Temperley-Lieb algebra.


Constructions

;Subalgebras: A subalgebra of an ''R''-algebra ''A'' is a subset of ''A'' which is both a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
and a
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
of ''A''. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of ''A''. ;Quotient algebras: Let ''A'' be an ''R''-algebra. Any ring-theoretic
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
''I'' in ''A'' is automatically an ''R''-module since ''r'' · ''x'' = (''r''1''A'')''x''. This gives the quotient ring ''A'' / ''I'' the structure of an ''R''-module and, in fact, an ''R''-algebra. It follows that any ring homomorphic image of ''A'' is also an ''R''-algebra. ;Direct products: The direct product of a family of ''R''-algebras is the ring-theoretic direct product. This becomes an ''R''-algebra with the obvious scalar multiplication. ;Free products: One can form a free product of ''R''-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of ''R''-algebras. ;Tensor products: The tensor product of two ''R''-algebras is also an ''R''-algebra in a natural way. See tensor product of algebras for more details. Given a commutative ring ''R'' and any ring ''A'' the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
''R'' ⊗Z ''A'' can be given the structure of an ''R''-algebra by defining ''r'' · (''s'' ⊗ ''a'') = (''rs'' ⊗ ''a''). The functor which sends ''A'' to ''R'' ⊗Z ''A'' is left adjoint to the functor which sends an ''R''-algebra to its underlying ring (forgetting the module structure). See also:
Change of rings In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. *f_* M = \operator ...
.


Separable algebra

Let ''A'' be an algebra over a commutative ring ''R''. Then the algebra ''A'' is a right module over A^e := A^ \otimes_R A with the action x \cdot (a \otimes b) = axb. Then, by definition, ''A'' is said to separable if the multiplication map A \otimes_R A \to A, \, x \otimes y \mapsto xy splits as an A^e-linear map, where A \otimes A is an A^e-module by (x \otimes y) \cdot (a \otimes b) = ax \otimes yb. Equivalently, A is separable if it is a
projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characteriz ...
over A^e; thus, the A^e-projective dimension of ''A'', sometimes called the bidimension of ''A'', measures the failure of separability.


Finite-dimensional algebra

Let ''A'' be a finite-dimensional algebra over a field ''k''. Then ''A'' is an Artinian ring.


Commutative case

As ''A'' is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field ''k''. Now, a reduced Artinian local ring is a field and thus the following are equivalent # A is separable. # A \otimes \overline is reduced, where \overline is some algebraic closure of ''k''. # A \otimes \overline = \overline^n for some ''n''. # \dim_k A is the number of k-algebra homomorphisms A \to \overline.


Noncommutative case

Since a
simple Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
is a (full) matrix ring over a division ring, if ''A'' is a simple algebra, then ''A'' is a (full) matrix algebra over a division algebra ''D'' over ''k''; i.e., A = M_n(D). More generally, if ''A'' is a semisimple algebra, then it is a finite product of matrix algebras (over various division ''k''-algebras), the fact known as the Artin–Wedderburn theorem. The fact that ''A'' is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of ''A'' is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.) The Wedderburn principal theorem states: for a finite-dimensional algebra ''A'' with a nilpotent ideal ''I'', if the projective dimension of A/I as an (A/I)^e-module is at most one, then the natural surjection p: A \to A/I splits; i.e., A contains a subalgebra B such that p, _B : B \overset\to A/I is an isomorphism. Taking ''I'' to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of
Levi's theorem In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semis ...
for
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s.


Lattices and orders

Let ''R'' be a Noetherian integral domain with field of fractions ''K'' (for example, they can be \mathbb, \mathbb). A '' lattice'' ''L'' in a finite-dimensional ''K''-vector space ''V'' is a finitely generated ''R''-submodule of ''V'' that spans ''V''; in other words, L \otimes_R K = V. Let A_K be a finite-dimensional ''K''-algebra. An ''
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
'' in A_K is an ''R''-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., \mathbb is a lattice in \mathbb but not an order (since it is not an algebra). A ''maximal order'' is an order that is maximal among all the orders.


Related concepts


Coalgebras

An associative algebra over ''K'' is given by a ''K''-vector space ''A'' endowed with a bilinear map ''A'' × ''A'' → ''A'' having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism ''K'' → ''A'' identifying the scalar multiples of the multiplicative identity. If the bilinear map ''A'' × ''A'' → ''A'' is reinterpreted as a linear map (i. e.,
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
in the category of ''K''-vector spaces) ''A'' ⊗ ''A'' → ''A'' (by the universal property of the tensor product), then we can view an associative algebra over ''K'' as a ''K''-vector space ''A'' endowed with two morphisms (one of the form ''A'' ⊗ ''A'' → ''A'' and one of the form ''K'' → ''A'') satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using
categorial duality In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the so ...
by reversing all arrows in the
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
s that describe the algebra
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s; this defines the structure of a coalgebra. There is also an abstract notion of ''F''-coalgebra, where ''F'' is a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
. This is vaguely related to the notion of coalgebra discussed above.


Representations

A representation of an algebra ''A'' is an algebra homomorphism ''ρ'' : ''A'' → End(''V'') from ''A'' to the endomorphism algebra of some vector space (or module) ''V''. The property of ''ρ'' being an algebra homomorphism means that ''ρ'' preserves the multiplicative operation (that is, ''ρ''(''xy'') = ''ρ''(''x'')''ρ''(''y'') for all ''x'' and ''y'' in ''A''), and that ''ρ'' sends the unit of ''A'' to the unit of End(''V'') (that is, to the identity endomorphism of ''V''). If ''A'' and ''B'' are two algebras, and ''ρ'' : ''A'' → End(''V'') and ''τ'' : ''B'' → End(''W'') are two representations, then there is a (canonical) representation ''A'' \otimes ''B'' → End(''V'' \otimes ''W'') of the tensor product algebra ''A \otimes B'' on the vector space ''V \otimes W''. However, there is no natural way of defining a
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by '' tensor product of representations'', the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a
Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Swe ...
or a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
, as demonstrated below.


Motivation for a Hopf algebra

Consider, for example, two representations \sigma:A\rightarrow \mathrm(V) and \tau:A\rightarrow \mathrm(W). One might try to form a tensor product representation \rho: x \mapsto \sigma(x) \otimes \tau(x) according to how it acts on the product vector space, so that :\rho(x)(v \otimes w) = (\sigma(x)(v)) \otimes (\tau(x)(w)). However, such a map would not be linear, since one would have :\rho(kx) = \sigma(kx) \otimes \tau(kx) = k\sigma(x) \otimes k\tau(x) = k^2 (\sigma(x) \otimes \tau(x)) = k^2 \rho(x) for ''k'' ∈ ''K''. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: ''A'' → ''A'' ⊗ ''A'', and defining the tensor product representation as :\rho = (\sigma\otimes \tau) \circ \Delta. Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A
Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Swe ...
is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).


Motivation for a Lie algebra

One can try to be more clever in defining a tensor product. Consider, for example, :x \mapsto \rho (x) = \sigma(x) \otimes \mbox_W + \mbox_V \otimes \tau(x) so that the action on the tensor product space is given by :\rho(x) (v \otimes w) = (\sigma(x) v)\otimes w + v \otimes (\tau(x) w) . This map is clearly linear in ''x'', and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication: :\rho(xy) = \sigma(x) \sigma(y) \otimes \mbox_W + \mbox_V \otimes \tau(x) \tau(y). But, in general, this does not equal :\rho(x)\rho(y) = \sigma(x) \sigma(y) \otimes \mbox_W + \sigma(x) \otimes \tau(y) + \sigma(y) \otimes \tau(x) + \mbox_V \otimes \tau(x) \tau(y). This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
.


Non-unital algebras

Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital. One example of a non-unital associative algebra is given by the set of all functions ''f'': R → R whose
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
as ''x'' nears infinity is zero. Another example is the vector space of continuous periodic functions, together with the convolution product.


See also

*
Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
*
Algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
*
Algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
*
Sheaf of algebras In algebraic geometry, a sheaf of algebras on a ringed space ''X'' is a sheaf of commutative rings on ''X'' that is also a sheaf of \mathcal_X-modules. It is quasi-coherent if it is so as a module. When ''X'' is a scheme, just like a ring, one ...
, a sort of an algebra over a
ringed space In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...


Notes


References

* * * * Nathan Jacobson, Structure of Rings * James Byrnie Shaw (1907
A Synopsis of Linear Associative Algebra
link from
Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
Historical Math Monographs. * Ross Street (1998)
Quantum Groups: an entrée to modern algebra
', an overview of index-free notation. * {{DEFAULTSORT:Associative Algebra Algebras Algebraic geometry